the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Pressuredependent performance of two CENspecified condensation particle counters
Paulus S. Bauer
Dorian Spät
Martina Eisenhut
Andreas Gattringer
Bernadett Weinzierl
One of the most important parameters to quantify an aerosol is the particle number concentration. Condensation particle counters (CPCs) are commonly used to measure the aerosol number concentration in the nanometer size range. To compare the data from different measurement stations and campaigns, it is important to harmonize the instrument specifications, which is why the CEN/TS 16976:2016 technical specification was introduced for CPCs. Therein, the parameters of the CEN CPC are specified for standard pressure and temperature. However, CEN CPCs are used in various surroundings, on high mountains or on airplanes, where they are exposed to lowpressure conditions. Here, we present the pressuredependent performance (including the concentration linearity and counting efficiency) of two different models of CEN CPCs, the Grimm 5410 CEN and the TSI 3772 CEN. We found that their performance at 1000 and 750 hPa was in accordance with the CEN technical specifications. Below 500 hPa, the performance decreased for both CPC models, but the decrease was different for the two models. To gain insight into the performance of the two CPC models, we performed a simulation study. This study included simulations of the saturation profiles and calculations of internal particle losses within the CPCs. The simulations reproduced the overall performance decrease with decreasing pressure and reveal that the internal structure of the CPC has a significant influence on the performance. We anticipate our publication to provide a deeper understanding of the counting efficiency of CPCs and their pressure dependence. Our findings might be a starting point for new standards that include the pressuredependent performance, or they could help in designing new CPCs.
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Atmospheric aerosol substantially influences human health (Oberdörster et al., 2005; WHO, 2016) and our climate (e.g., IPCC, 2021, 2013). Therefore, it is constantly monitored either by groundbased measurement stations (such as Aerosol, Clouds and Trace Gases Research Infrastructure (ACTRIS) and Global Atmosphere Watch (GAW); Asmi et al., 2013; Rose et al., 2021) or by aircraft measurements (IAGOS, the ATOM mission, ALIFE, and many other ones; Bundke et al., 2015; Williamson et al., 2018; Brock et al., 2019; Kupc et al., 2018; Weinzierl et al., 2017; Thompson et al., 2022; Schöberl et al., 2023). The particle number concentration is an important parameter for quantifying the abundance of these shortlived atmospheric components. Condensation particle counters (CPCs) are commonly used to assess the number concentration directly in the nanometer size range. CPCs are operated under various environmental conditions including measurements at alpine monitoring stations, e.g., Sonnblick Observatory (SBO; Rose et al., 2021) or the Himalayas (Bianchi et al., 2020), and in aircraft (Bundke et al., 2015; Williamson et al., 2018; Brock et al., 2019; Kupc et al., 2018). Due to the decreased ambient pressure at these altitudes, the instruments' performance might change. Since more than 30 years the pressuredependent performance of CPCs is evaluated in numerous studies (e.g., Hermann and Wiedensohler, 2001; Zhang and Liu, 1991, 1990; Schröder and Ström, 1997; Heintzenberg and Ogren, 1985; Noone and Hansson, 1990; Dreiling and Jaenicke, 1988; Cofer et al., 1998). Most of these studies have used butanolbased CPCs, but there are also studies with different working fluids (e.g., Hermann et al., 2005; Bezantakos and Biskos, 2021; Mei et al., 2021; Weigel et al., 2009) or with different analysis and correction methods (e.g., Takegawa and Sakurai, 2011; Seifert et al., 2004; Saros et al., 1996) to account for performance differences with varying pressure.
However, these studies evaluated mostly one CPC type with individual performances. To compare and evaluate the data from different monitoring stations and measurement campaigns, it is important to harmonize the instrument specifications. The CEN/TS 16976:2016 technical specification (CEN/TS 16976, 2016, published in 2016 and currently valid but changes expected with the next revision) defines the CPC specifications at standard pressure and temperature (details in Sect. 1.2). It is therefore possible to directly compare the performance of different CEN CPC models at ambient conditions but also under varying pressure conditions. Thus, the presented study aims at assessing the pressuredependent performance of two models of CENCPCs, namely the Grimm 5410 CEN (in the following referred to as Grimm CPC) and the TSI 3772 CEN (in the following referred to as TSI CPC) (details on the CPCs in Sect. 1.2).
1.1 The counting efficiency of condensation particle counters
Condensation particle counters represent one of the oldest measurement techniques in aerosol science, and various different types of CPCs have been developed (McMurry, 2000). In this study we focused on continuousflow thermaldiffusiontype CPCs with an alcoholbased working fluid (McMurry, 2000). This means that the aerosol is continuously sampled by a CPC, consisting of a saturator, a condenser, and a detection unit. For alcoholbased CPCs, the saturator is operated at a higher temperature compared to the condenser, since the thermal diffusion rate (heat transfer) in air is higher than the diffusion rate of the (relatively large) alcohol molecules (Iida et al., 2009; McMurry, 2000; Hering and Stolzenburg, 2005). In the warm saturator, the working fluid is vaporized and saturates the aerosol. Afterwards the aerosol enters the cold condenser where supersaturation is generated by thermal and vapor diffusion due to the rapid temperature change. Depending on the supersaturation, particle size, and chemical composition, particles are activated by heterogeneous nucleation. By subsequent condensational growth, the activated particles grow to sizes large enough to be optically counted (e.g., with a laser) in the detector unit.
One important parameter to quantify the performance of a CPC is the counting efficiency η_{CPC} as a function of the particle size d_{p}. According to Stolzenburg and McMurry (1991), the CPC counting efficiency can be decomposed into
where η_{s} is the sampling efficiency, η_{a} the activation efficiency, and η_{d} the detector efficiency inside the CPC. The sampling efficiency η_{s} is determined by the sampling and transport losses inside the CPC. The activation efficiency η_{a} accounts for the fraction of particles that are activated by heterogeneous nucleation in the condenser. Here it should be pointed out that the activation efficiency does not only depend on particle size but also on the interactions between the particle and the vapor, e.g., solubility and wettability (Kupc et al., 2013). These chemicaldependent variables are not considered in this study but are discussed in other publications (e.g., Wlasits et al., 2020; Giechaskiel et al., 2011; Köhler, 1936). The detector efficiency η_{d} comprises all activated particles that grow to droplets large enough to be measured by the optical detection system. In the original publication by Stolzenburg and McMurry (1991) the “detector efficiency η_{d}” is named “detection efficiency”; however we renamed it to avoid confusion with the term detection efficiency, which is sometimes used synonymously for the counting efficiency η_{CPC}. In general, these three different efficiencies inside the CPC cannot be measured individually but can only be evaluated theoretically using simulations.
The experimental approach to measure the counting efficiency ${\mathit{\eta}}_{\mathrm{CPC}}^{\prime}$ is to generate a monodisperse aerosol with a defined particle size d_{p} (conventionally using a differential mobility analyzer, DMA; multiple charged particles have to be taken into account). For different particle sizes, the detected concentration of the CPC, N_{CPC}, is then compared to the concentration of a reference instrument N_{REF} (conventionally a Faraday cup electrometer, FCE):
The measured concentrations (N_{CPC} and N_{REF}) also contain the transport efficiencies to the instruments and the size distribution of the aerosol (i.a. DMA transfer function), which must be considered for the exact counting efficiency (details in Stolzenburg and McMurry, 1991). Several measures were taken to minimize these effects described in the Methods section. As a result, the experimental counting efficiency is calculated as stated in Eq. (2) in this study.
The counting efficiency curve η(d_{p}) has a very specific form with an increasing slope and a plateau region going from the small to the large particle sizes (see Fig. 3 for an example). It can be described by three important parameters, which are the plateau counting efficiency η_{plat}, the cutoff diameter d_{p,50}, and the onset diameter d_{p,0}. Going from the largest sizes to the smallest, the plateau counting efficiency ${\mathit{\eta}}_{\mathrm{plat}}=\mathit{\eta}({d}_{\mathrm{p},\mathrm{50}}\ll {d}_{\mathrm{p}}<\mathrm{1}$ µm) is the counting efficiency at large particle sizes (at least much larger than the cutoff diameter d_{p,50}; see next parameter) and is ideally η_{plat}=100 % for the CENspecified ambient conditions (see Sect. 1.2). The cutoff diameter d_{p,50} is the diameter where the counting efficiency reaches 50 %, $\mathit{\eta}\left({d}_{\mathrm{p},\mathrm{50}}\right)=\mathrm{50}$ %. The cutoff diameter is conventionally seen as the lower detection limit of the measurement range of a CPC. The third parameter is the onset diameter d_{p,0}, which is the diameter of the smallest particles that are detected by the CPC, $\mathit{\eta}\left({d}_{\mathrm{p},\mathrm{0}}\right)\approx \mathrm{0}$ %. Since the onset diameter cannot be obtained directly with these types of measurements, simulations or a fit function are utilized. The fit function including these three parameters is (Tuch et al., 2016; Stolzenburg and McMurry, 1991; ISO27891, 2015)
It should be pointed out that the fitted cutoff diameter d_{p,50fit} ($\mathit{\eta}\left({d}_{\mathrm{p},\mathrm{50}\mathrm{fit}}\right)/{\mathit{\eta}}_{\mathrm{plat}}=\mathrm{50}$ %) is only equal to the previously defined cutoff diameter d_{p,50} if the plateau counting efficiency is η_{plat}=100 %. The fitted cutoff diameter d_{p,50fit} is shifted to lower diameters compared to the cutoff diameter d_{p,50} if the plateau counting efficiency η_{plat} is lower than 100 %.
To describe the increasing slope of the counting efficiency curve η(d_{p}), we introduce the edge steepness parameter ε. Ideally the steepness is the derivative of the counting efficiency curve $\mathrm{d}\mathit{\eta}\left({d}_{\mathrm{p}}\right)/\mathrm{d}{d}_{\mathrm{p}}$ in the slope region. We approximate this derivative with the difference quotient between the onset and the cutoff diameter. We therefore define the edge steepness parameter as
where d_{p,0} is the onset diameter derived from the fit, and d_{p,50} is the cutoff diameter. In principle, the edged steepness could be calculated with the fitted cutoff diameter d_{p,50fit}. However, d_{p,50fit} depends on the plateau counting efficiency η_{plat}, which is why we will not use d_{p,50fit} for the edge steepness in this publication. In general, the edge steepness ε represents the percentual increase in the counting efficiency per nanometer of particle size between 0 % and 50 %, and thus a steeper slope will give a larger edge steepness ε.
1.2 CEN 16976:2016 technical specification
The CEN/TS 16976:2016 technical specification, entitled “Determination of the particle number concentration of atmospheric aerosol”, was published in 2016. It covers a variety of related topics, ranging from the sampling inlet system to the specifications of the counting devices, which in this case is a CPC. Some specifications are based on ISO27891:2015 (“Aerosol particle number concentration – Calibration of condensation particle counters”), published in 2015. CEN/TS 16976 will become a European standard in the near future, and some parameters (e.g., the cutoff diameter d_{p,50} or pressure calibration) might change. However, when the publication was written, CEN/TS 16976:2016 was still valid, and the most relevant specifications for the presented study are summarized in the following paragraphs.
The CPC has to be a fullflow CPC, which means that there is no internal dilution of the aerosol flow inside the CPC. The volumetric flow rate should only deviate by 5 % from the nominal volumetric flow rate. The working fluid has to be nbutanol. Silver nanoparticles generated by the evaporation–condensation method (Scheibel and Porstendörfer, 1983) have to be used for the verification of the counting efficiency. The particle concentration should be between 3000 and 10 000 cm^{−3}. However, it is not clearly specified if that concentration is determined using the CPC or the reference instrument and which carrier gas should be used (air or N_{2}). The linearity of the concentration of the CPC in the plateau region has to be measured at one fixed particle diameter between 30 and 50 nm by varying the concentration. The linearity (and hence the plateau counting efficiency η_{plat}) has to be 1 ± 5 %. The cutoff diameter has to be determined by a fit, Eq. (3), and d_{p,50fit} should be 7 ± 0.7 nm. Since the plateau counting efficiency should be nearly 1, the fitted d_{p,50fit} is almost equal to the cutoff d_{p,50}. However, for some of the following measurements, at low pressure the plateau is lower than 1, and thus both cutoff diameter parameters (d_{p,50} and d_{p,50fit}) have to be considered.
In addition to the cutoff diameter, the diameter corresponding to a counting efficiency of 90 % d_{p,90} should be below 14 nm. The measurements of the counting efficiency should be done at two different temperatures (15 and 30 ^{∘}C) and at two different pressures, one higher than 900 hPa and one 200 hPa lower compared to an unknown reference. At this point, the technical specification is vague as it lacks a reference for the lowerpressure measurement and does not contain specifications for a setup capable of creating these lowpressure conditions.
The standard temperature T_{0} = 296.15 K, the standard pressure p_{0} = 1013.25 hPa, the equations for the mean free path λ_{air}(T,p), the dynamic viscosity μ(T), and the Cunningham correction factor C_{c}(d_{p},λ_{air}) are specified in CEN/TS 16976 and can be found in Wiedensohler et al. (2012).
CENspecified condensation particle counters
We tested two different models of CENspecified CPCs: the Grimm 5410 CEN and the TSI 3772 CEN. Here the characteristics of both CPC models are presented in alphabetical order. The parameters of the two CPCs are presented in Table 1. Both CPC models are specified as fullflow CPCs with nbutanol as the working fluid, and they require an external vacuum pump.
The Grimm 5410 CEN has a nominal flow rate of 0.6 L min^{−1} controlled by a temperaturestabilized critical orifice. The saturator temperature is T_{sat} = 36 ^{∘}C, and the condenser temperature is T_{con} = 17 ^{∘}C. The Grimm saturator has a displacer rod in the center, which must be considered to be an annular tube in simulations and particle loss calculations. Singleparticle counting is possible up to 10^{5} cm^{−3} with internal coincidence correction for the Grimm CPC (Manual Grimm 5410; Grimm, 2020).
The TSI 3772 CEN has a nominal flow rate of 1 L min^{−1} controlled by a critical orifice. The flow is internally split up into eight pathways, each with a flow rate of 0.125 L min^{−1} for the condenser and saturator part (Kangasluoma et al., 2014). The saturator temperature is T_{sat} = 39 ^{∘}C, and the condenser temperature is T_{con} = 18 ^{∘}C. Singleparticle counting is possible up to 5 × 10^{4} cm^{−3} with livetime coincidence correction for the TSI CPC (Manual TSI 3772 CEN; TSI, 2016).
2.1 Experimental setup
Figure 1 shows a schematic of the setup that we used to characterize the performance of the CEN CPCs under lowpressure conditions. According to the CEN/TS 16976:2016, silver nanoparticles were generated via the evaporation–condensation method (Scheibel and Porstendörfer, 1983). Silver was heated between 940 and 1050 ^{∘}C in a tube furnace and subsequently cooled with a Liebig water cooler (15 ^{∘}C). A dilution flow was added to adjust the number concentration and size distribution of the nanoparticles. The furnace flow and the dilution flow were operated with laboratorypressurized air. Each flow was equipped with a needle valve, a silicagel dryer, a HEPA filter, and a mass flow meter (TSI 4140) for precise flow control. The relative humidity of the air supply was kept below 10 %.
The particles were selected corresponding to their electrical mobility with a classification system consisting of a soft Xray charger (TSI 3087) and a custommade Viennatype differential mobility analyzer (DMA; presented in Winkler et al., 2008b, and Wlasits et al., 2020, referred to as a nanoDMA). A positive voltage was applied to select negatively charged particles with an equivalent mobility diameter ranging from 4 to 30 nm. A sheath air flow of 25 L min^{−1} was generated with a closedloop flow system, including HEPA filters, a silicagel dryer, a critical orifice, and a pump. The flow through the classification system (often referred to as Q_{a}, the aerosol flow, or Q_{s}, the sample flow) was determined by the flow of the sampling system (mostly the flow through the limiting orifice). The resolution of the DMA defined by the flow ratio (Flagan, 1999) was therefore nearly constant at 1 : 10 for all measurements. Diffusional broadening inside the DMA was not considered as the relevant mobility diameters were above 5 nm (Wlasits et al., 2020). Multiple charged particles were considered as described in the “Experimental procedure”, Sect. 2.2.
For lowpressure measurements, particles must either be sizeselected (mobilityselected) in the lowpressure region (Hermann and Wiedensohler, 2001), or the monodisperse particles must be transferred into the lowpressure region via a valve (Zhang and Liu, 1991) or an orifice (Takegawa and Sakurai, 2011). We tested the valve and the orifice system, both resulting in similar counting efficiencies (not shown in this publication). However, the setup with the orifice similar to Takegawa and Sakurai (2011) yielded higher particle concentrations and better control regarding concentration and pressure; hence we used this setup. We did not see any charging artifacts in either system (see discussion in Hermann and Wiedensohler, 2001, and Takegawa and Sakurai, 2011). The green part in Fig. 1 indicates where the relevant pressure conditions are. For measurements at ambient pressure, the critical orifice was replaced by a stainlesssteel pipe and the sampling flow was adjusted to 2.0 ± 0.1 L min^{−1}. At our measurement location in Vienna, the average ambient pressure was 996 ± 15 hPa during our measurement period, which is why we label all ambient pressure stages as 1000 hPa. At the 750 hPa pressure stage (after the orifice), we measured a flow rate of 1.9 ± 0.1 L min^{−1} (in front of the orifice). For the pressure stages at 500 hPa and below, a critical pressure ratio ${p}_{\mathrm{after}}/{p}_{\mathrm{before}}<\mathrm{0.528}$ (pressure after and before the orifice) (Wiggert et al., 2016; Rathakrishnan, 2017) was sustained, ensuring a stable flow of 2.1 L min^{−1}.
After the orifice, an aluminum mixing chamber assured that the aerosol was relaxed and well mixed before the splitter. The “adjustment flow”, which exits the mixing chamber halfway, controls the pressure stages. The stainless steel Y splitter guaranteed equal splitting of the aerosol flow for the detection instruments, which was verified by swapping the instruments from one outlet to the other. From the DMA to the Y splitter, every part and connection consisted of metal, and the instrumentation was attached with conductive tubing of equal length to reduce electrostatic deposition and ensure similar transport losses for all instruments.
As a reference instrument we used a Faraday cup electrometer (FCE) from TAPCON (Winkler et al., 2008b, a). The volumetric flow rate of the FCE was set to the nominal flow rate of the CPC (Q_{Grimm} = 0.6 L min^{−1}, Q_{TSI} = 1 L min^{−1}) using an Alicat mass flow controller (MC series). The pressure sensor (precision ±0.1 %) of the mass flow controller was also used as a reference instrument for our pressure stages (green part in Fig. 1). A pressure gauge [P] (Jumo Delos SI, precision ±0.35 %) measured the pressure before the Agilent pump, which was max. 50 hPa. This assured that the flow of the CPC's critical orifice stayed choked (${p}_{\mathrm{after},\mathrm{CPC}}/{p}_{\mathrm{before},\mathrm{CPC}}<\mathrm{0.528}$; Wiggert et al., 2016; Rathakrishnan, 2017) for all pressure stages down to 150 hPa. With the Jumo pressure gauge, we also verified the pressure stability at various points between the limiting orifice and the instrumentation, which was consistent with the pressure reading of the mass flow controller.
2.2 Experimental procedure
Before we started with the counting efficiency measurements, we analyzed the size distribution of the silver nanoparticles by operating the DMA in scanning mode. We adjusted the temperatures and flows of the furnace to produce a minimum particle concentration of 2000 cm^{−3} and a maximum of 20 000 cm^{−3} measured by the FCE in the mobility size range between 4 and 30 nm. CEN/TS 16976:2016 restricts the maximum particle concentration to 10 000 cm^{−3}; however it is not clearly specified whether this maximum concentration applies to the CPC or the FCE. Furthermore, both CPCs are built to measure particle concentrations of up to 50 000 cm^{−3} (TSI 3772 CEN) or even beyond (Grimm 5410 CEN), when coincidence correction is switched on. Both CPC models come with an internal coincidence correction (e.g., “livetime correction” for TSI 3772 CEN), which is why we chose the corrected concentration output of the CPCs (N_{CPC}) for our data analysis.
To avoid multiply charged particles, the size distribution was adjusted so that the mode of the distribution was lower than 30 nm and the concentration at 30 nm was on the lower end (slightly above 2000 cm^{−3}). For the linear response and concentration comparison between CPC and FCE, CEN/TS 16976:2016 recommends mobility particle sizes of 40 ± 10 nm. We used 30 nm particles selected from the right flank of the size distribution, where they can be considered to be singly charged particles (Tuch et al., 2016; Wiedensohler, 1988).
The experimental procedure for the counting efficiency measurements was automated and always started with a 2 min zero measurement (0 V at the DMA) to set the reference for the FCE and check the zero counts of the CPC. Then we alternately set 0 V for 1 min and the voltage for the desired mobility diameters for 2 min at the DMA. The data (about 10 to 20 s) before and after each voltage transition were removed before taking the average because of spikes in the concentration when the voltage was ramped up or down (Takegawa and Sakurai, 2011). For the FCE, we took the mean of the zero measurements before and after the 2 min interval and subtracted the result from the mean of the 2 min interval to correct for the FCE background. The uncertainties resulting from this procedure were analyzed with Gaussian error propagation. To account for daytoday variations, we started each measurement day with a counting efficiency measurement at ambient pressure (labeled as 1000 hPa) and checked that it was consistent with previous measurements with instruments of the same type. In addition, we checked the flow rate of the sheath air flow (in and out of the DMA) and the flow before and after the DMA each time the flow or the pressure level was changed.
For the lowpressure measurements, the butanol supply of the CPCs was removed, and the autofill mode was switched off to avoid pressure leakage or flooding of the CPC. The stability of the counting efficiency without butanol supply was checked for both CPC models by monitoring the efficiency of 30 nm particles over a long period of several hours, similar to Takegawa and Sakurai (2011). Even for the lowest pressure settings where butanol diffusion and hence butanol losses are the largest, we were able to measure more than 6 h without any change in the counting efficiency. Despite these results, we filled up the CPCs with butanol after each counting efficiency measurement routine under low pressure to assure equal conditions for each measurement.
2.3 Simulation methods
A simulation of the CPC is needed to investigate the individual efficiencies from Eq. (1) leading to the total counting efficiency measured experimentally. There are many publications simulating the condenser or the whole CPC to investigate these efficiencies (e.g., Stolzenburg and McMurry, 1991; Zhang and Liu, 1990; Hering and Stolzenburg, 2005; Giechaskiel et al., 2011; Reinisch et al., 2019). The heart of these simulations is the equations for heat and mass transfer to calculate the temperature and vapor pressure profiles inside the CPC. The temperature profiles from a tube, where the wall temperature makes a sudden step change similar to in a CPC, has been known in literature since the end of the 19th century as the Graetz–Nusselt problem (Eckert and Drake, 1972; Bird et al., 2002). The solution to the Graetz–Nusselt problem is often used as a reference for the simulations (e.g., Giechaskiel et al., 2011; Reinisch et al., 2019).
For our simulations we have made several assumptions: the problem is cylindrical symmetric, so we consider only the axial (z) and radial (r) direction. We normalized the axial ${z}^{\prime}=z/{R}_{\mathrm{t}}$ and radial ${r}^{\prime}=r/{R}_{\mathrm{t}}$ distance with the radius of the tube R_{t}. The flow is incompressible and laminar and has a fully developed parabolic flow profile $v\left({r}^{\prime}\right)=\mathrm{2}\stackrel{\mathrm{\u203e}}{v}(\mathrm{1}{r}^{\prime \mathrm{2}}$), where $\stackrel{\mathrm{\u203e}}{v}=Q/\mathit{\pi}{R}_{\mathrm{t}}^{\mathrm{2}}$ is the average velocity of the flow and Q the volumetric flow rate (m^{3} s^{−1}). We do not consider any diffusion in axial direction nor coupling effects between mass and thermal diffusion, nor effects from Stefan flow (Stolzenburg and McMurry, 1991). Heat and mass transfer onto the growing droplets is negligible (Zhang and Liu, 1990), and particle–particle interactions are neglected (Stolzenburg and McMurry, 1991). Thus, the results from the simulation are most accurate for monodisperse aerosol with a low particle concentration to neglect vapor depletion effects.
With these assumptions, the equations for heat and mass transfer can be reduced to
where Pe is the Péclet number, and ψ is a placeholder variable that could either be the temperature T or the partial vapor pressure p_{v} for the heat and mass transfer equations, respectively. The derivation of Eq. (5) can be found in the Supplement or partly in Bird et al. (2002). To solve this partial differential equation, we used the FEniCS computer platform (Alnæs et al., 2015).
The physical properties of the gas, the butanol vapor, and the operating parameters of the CPC are incorporated into the Péclet number. The Péclet number Pe is a dimensionless number comparing the advective and the diffusive transport rate. For thermal processes, the Péclet number Pe_{T} can be decomposed into the dimensionless Reynolds Re and Prandtl number Pr:
where ν (m^{2} s^{−1}) is the kinematic viscosity ($\mathit{\nu}=\mathit{\mu}/\mathit{\rho}$), μ (Pa s) the dynamic viscosity, ρ the density of the gas (kg m^{−3}), α (m^{2} s^{−1}) the thermal diffusivity ($\mathit{\alpha}={k}_{\mathrm{T}}/(\mathit{\rho}\cdot {c}_{p})$), k_{T} (W m^{−1} K^{−1}) the thermal conductivity, and c_{p} (J kg^{−1} K^{−1}) the specific heat capacity at constant pressure. For the partial vapor pressure, the Péclet number Pe${}_{{p}_{\mathrm{v}}}$ can be decomposed into the dimensionless Reynolds Re and Schmidt number Sc:
where D_{v} (m^{2} s^{−1}) is the (binary) diffusion constant of the vapor in air. In Zhang and Liu (1990), it is shown that the Prandtl number Pr and the Schmidt number Sc only depend on temperature and not on pressure. However, the Reynolds number can be written as
where Re_{0}, Q_{0}, and p_{0} are the Reynolds number, the volumetric flow rate, and the pressure at standard operation conditions. This implies that either a reduction of the pressure by some factor or a reduction of the volumetric flow rate by the same factor results in the same heat and mass transfer equations (Zhang and Liu, 1990). We will only focus on the pressure dependence in this publication.
To solve the partial differential equation (Eq. 5), several boundary conditions are necessary. In most CPC simulation studies (e.g., Zhang and Liu, 1990; Hering and Stolzenburg, 2005; Giechaskiel et al., 2011), only the condenser of the CPC is simulated. However, we included the insulator between the saturator and condenser in our simulations similar to Reinisch et al. (2019), which is of importance especially for the lowpressure cases. In the insulator, the wall temperature T_{wall} is linearly decreasing from the saturator temperature T_{sat} to the condenser temperature T_{con}. In the condenser the wall temperature T_{wall} is constant at the condenser temperature T_{con}. The partial vapor at the wall p_{wall}=p_{sat}(T_{wall}) is set to the saturation vapor pressure at wall temperature. The incoming aerosol has the temperature of the saturator T_{sat} and is considered to be fully saturated with butanol vapor (Reinisch et al., 2019). This is especially true for the lowpressure case because the molecular diffusion is enhanced if the pressure is reduced, and hence the aerosol gets saturated more easily. The vapor pressure of the incoming aerosol p_{v}=p_{sat}(T_{sat}) is set to the saturation vapor pressure at saturator temperature.
With the resulting temperature T and partial vapor pressure p_{v} profiles, we calculated the saturation ratio profiles, $S={p}_{\mathrm{v}}/{p}_{\mathrm{sat}}\left(T\right)$, where p_{sat}(T) is the saturation vapor pressure at temperature T. Each point of the saturation ratio profile S and temperature profile T can be linked to an equilibrium diameter D_{K,eq} via Kelvin theory:
where σ_{s}, M_{w}, and ρ_{l} are the surface tension, the molecular weight, and the density of the condensing fluid (in our case liquid butanol); R is the universal gas constant; and T is the absolute temperature (Winkler and Wagner, 2022). This socalled Kelvin diameter represents the minimal particle size that gets activated in the conditions present around the particle.
To calculate the activation efficiency η_{a} from the simulations, we discretized the profiles into axial K_{ax} and radial K_{rad} bins. Then the Kelvin diameter was calculated for each bin. For each particle size d_{p} and for each radial bin (with the normalized radius ${r}_{i}^{\prime}$), we determined if and where (in axial direction) the particle gets first activated. With this information, we calculated the concentration of activated particles N_{act}(d_{p},r_{i}) for each particle size and radial bin, which we compared to the incoming number concentration N_{in}(d_{p},r_{i}). We then computed the activation efficiency η_{a} with (Giechaskiel et al., 2011; Reinisch et al., 2019)
Here the factor $(\mathrm{1}{r}_{i}^{\prime \mathrm{2}}$) accounts for the flow profile and the factor ${r}_{i}^{\prime}$ for the increase in the bin size and hence particle number concentration for each bin with the radial position ${r}_{i}^{\prime}$ (Reinisch et al., 2019).
To investigate the total counting efficiency of Eq. (1), we examined the sampling efficiency η_{s}, which includes the particle losses from the inlet of the CPC to its condenser. For the size range below 100 nm we only considered diffusional losses, and therefore we implemented Eqs. (21) and (22) from von der Weiden et al. (2009) for cylindrical tubes. For the particle losses in the annular saturator of the Grimm CPC, we used the formula from Talebizadehsardari et al. (2020). The detector efficiency η_{d} includes the growth of the particles to optical sizes, which we have analyzed with a growth model (in the Supplement). To conclude, all activated particles can be considered optically detectable due to the rapid growth (details in Hering and Stolzenburg, 2005; Giechaskiel et al., 2011). Neither losses in the focusing region nor in the optic section of the CPC were studied in this publication, and thus we set the detector efficiency η_{d}=1.
3.1 Experimental results
We analyzed four identically constructed Grimm 5410 CEN CPCs and two identically constructed TSI 3772 CEN CPCs. We conducted measurements at the pressure levels of 1000, 750, 500, 375 (only TSI CPCs), 250, and 150 hPa (only Grimm CPCs). In the first set of experiments, we analyzed the linearity of the CPCs compared to the reference FCE, shown in Fig. 2. In a second set of experiments, the sizedependent counting efficiency curves for different pressure stages are shown in Fig. 3. The results of the fits performed in Figs. 2 and 3 are reported in Table 2. For clarity, Figs. 2 and 3 and Table 2 are presented without error bars or uncertainties, but detailed figures and tables with the necessary information are available in the Supplement.
The linear response of the CPC is very important for the later counting efficiency measurements. We analyzed the linearity at 30 nm for one CPC of each CPC model. In Fig. 2 the concentration of the CPCs (with coincidence correction switched on) is compared to the reference concentration of the FCE for the different pressure stages in a log–log plot. The results of the linear fit through the origin are reported in the CPC $/$ FCE column in Table 2. For clarity the 750 hPa points and fits are not shown in Fig. 2 but are plotted in the Supplement.
At 1000 and 750 hPa, the linearity of both CPC models is in agreement with the CEN specifications (1 ± 5 %; see Sect. 1.2). At 500 and 250 hPa, the response of both CPC models was still linear but below 1. This linear behavior (below 2 × 10^{4} cm^{−3}) prompted us to utilize the coincidencecorrected concentration of the CPCs for the counting efficiency measurements. Interestingly, at 500 hPa, the linear fit results for both CPC models are around 93 %, whereas at 250 hPa, the Grimm CPC is still above 80 % and the TSI CPC is around 55 %. Therefore, we have added the 375 hPa pressure stage for the TSI CPC and the 150 hPa for the Grimm CPC to investigate this drop in plateau counting efficiency. Since the Grimm CPC showed some nonlinear behavior at 150 hPa for particle concentrations above 6000 cm^{−3}, we only considered FCE concentrations below 6000 cm^{−3} for the linear fit.
The sizedependent counting efficiency measurements were conducted for four Grimm CPCs and two TSI CPCs at different pressure stages, except for 150 hPa, which was only measured for three different Grimm CPCs. The resulting counting efficiency curves were very consistent between the different CPCs of each CPC model (see Supplement). Thus, we averaged the counting efficiency measurements for each CPC model, which is shown in Fig. 3. The counting efficiency curves were fitted with Eq. (3). The resulting parameters η_{plat}, d_{p,0}, and d_{p,50fit} are reported in Table 2, along with the subsequently obtained parameters d_{p,50} and d_{p,90} (see Sect. 1.1). The edge steepness ε calculated with Eq. (4) was added to quantify the slope of the counting efficiency curves. A higher value of the edge steepness ε represents a steeper slope.
The results from the linearity fit of Fig. 2 (CPC $/$ FCE in Table 2) and the results of η_{plat} from the counting efficiency fit of Fig. 3 (η_{plat} in Table 2) represent a measure of the plateau counting efficiency. Both representations of the plateau counting efficiency agree to each other within 5 % for pressures down to 250 hPa for the Grimm CPCs and down to 375 hPa for the TSI CPCs. This is notable since the plateau counting efficiency is determined in two different ways, with different fitting functions and procedures. In addition, the linearity of Fig. 2 is only measured with one CPC, whereas the data of Fig. 3 are the average of an ensemble of CPCs. Below the mentioned pressure stages, the plateau counting efficiencies (CPC $/$ FCE, η_{plat}) of both CPC models show a bigger difference, which originates from various effects including the nonlinear behavior of the CPCs, the averaging, and the flat shape (large edge steepness ε) of the counting efficiency fit in Fig. 3, which shifts η_{plat} to higher values. In general, there is a big change in the counting efficiency curves and fit parameters (see Fig. 3 and Table 2) from the respective secondlowest pressure stage (250 hPa for the Grimm CPCs and 375 hPa for the TSI CPCs) to the lowest pressure stage (150 hPa for the Grimm CPCs and 250 hPa for the TSI CPCs) for both CPC models.
Both CPC models show a similar trend regarding the pressure dependence of the counting efficiency, which is comparable to Takegawa and Sakurai (2011) and Zhang and Liu (1990). For decreasing pressure, the plateau counting efficiency η_{plat} is decreasing, the cutoff diameter (d_{p,50fit} and d_{p,50}) is increasing, and the edge steepness ε is decreasing (the curves are getting flatter). For 1000 and 750 hPa, the Grimm CPCs and the TSI CPCs are in agreement with the CEN technical specifications (see Sect. 1.2). At 500 hPa and below, the Grimm CPCs generally have a higher plateau counting efficiency and a lower cutoff diameter than the TSI CPCs. A further difference between the CPC models is the pressure dependence of the onset diameter d_{p,0}. For the Grimm CPCs the onset diameter changes only marginally with pressure, which is why the curves in Fig. 3 seem to emerge from one point. For the TSI CPCs the onset diameter is increasing with decreasing pressure, which is why the curves look more separated. To investigate this difference in behavior between both CEN CPC models we conducted simulations, which are presented in the next section.
3.2 Numerical results
We simulated the temperature and the vapor pressure profile for the insulator and condenser of both CPC models as described in Sect. 2.3. From the results we calculated the saturation ratio S, depicted as contour plots in Fig. 4 for 1000 and 250 hPa. The corresponding centerline profiles (at r=0) of the saturation ratio S, the saturation vapor pressure, and the partial vapor pressure for 1000 and 250 hPa are displayed in Fig. 5. The simulations were performed for all pressure stages of the experimental results (Sect. 3.1), but for clarity only the profiles for 1000 and 250 hPa are provided. In all plots, the x axis represents the normalized length ${z}^{\prime}=z/{R}_{\mathrm{t}}$ (see Sect. 2.3). To visualize the different lengths of the CPC model's insulator and condenser the x axes are set to the same scale. The length of the insulator is indicated either with a vertical dashed line (Fig. 4) or with a gray shaded part (Fig. 5). The black 7 nm line in Fig. 4 encloses the area where the supersaturation is sufficient to activate at least 7 nm particles (Eq. 9). The black 7 nm line corresponds to the desired cutoff diameter d_{p,50} = 7 nm of the CEN specification (see Sect. 1.2).
From the simulated saturation ratio profiles, we calculated the activation efficiency η_{a} (Eq. 10 in Sect. 2.3) for the different pressure stages shown in Fig. 6. Combined with the sampling efficiency η_{s}, which includes the particle losses of the inlet and saturator of the CPC, we obtain the numerically calculated counting efficiencies presented in Fig. 7. In both figures the measured counting efficiencies of Fig. 3 were added as a reference. Table 3 presents the parameters evaluated from the numerically calculated counting efficiency (Fig. 7), the counting efficiency at d_{p}=30 nm (η(30 nm)), and the other parameters specified in Sect. 1.1 (onset d_{p,0}, cutoff d_{p,50}, 90 % diameter d_{p,90}, and edge steepness ε).
The profiles (Figs. 4 and 5) and the corresponding parameters give insights into the behavior of the different CPC models. One of the most distinct differences between the two CPC models is the relative lengths of the insulator and condenser. The Grimm CPC has a shorter insulator and a longer condenser than the TSI CPC relative to the tube radius. But, the TSI CPC has a lower flow rate in this section of the instrument (see Table 1), which has to be considered for the saturation profile S (see Eq. 8 in Sect. 2.3). First, we compare the saturation profiles of both CPC models at 1000 hPa. One important characteristic is the point with the highest (super) saturation ratio S_{max}, which is relevant for the smallest particles that get activated (see Kelvin equation (Eq. 9)) classified by the onset diameter d_{p,0}. For the Grimm CPC, S_{max} is located almost at the end of the condenser, whereas for the TSI CPC, it is close to the entrance of the condenser. For the TSI CPC, the saturation profile (especially the black 7 nm line) and vapor pressure profile reach significantly into the insulator. This demonstrates the importance of including the insulator in simulations, as stated in Reinisch et al. (2019).
At 250 hPa the saturation profile and S_{max} of both CPC models are shifted to the left (towards the entrance). This is in good agreement with theoretical considerations of Eq. (8) in Sect. 2.3. For the Grimm CPC, S_{max} is almost as high as at 1000 hPa, which only results in a small shift of the onset diameter (see Figs. 6 and 7). For the TSI CPC at 250 hPa the saturation profile is substantially different compared to the 1000 hPa. It is moved into the insulator part, and S_{max} is even lower than required for activating 7 nm particles (no black 7 nm line). Thus, the onset diameter of the TSI CPC is beyond 7 nm at 250 hPa. The onset diameter shift also explains why the activation efficiency curves η_{a} in Fig. 6 nearly fall onto each other for the Grimm CPC, and for the TSI CPC the curves and onset diameters are shifted for each pressure stage.
In Fig. 6, for all pressure stages the calculated activation efficiency η_{a} reaches 100 % at some particle diameter. The required supersaturation to activate a particle is decreasing exponentially with increasing Kelvin diameter (Eq. 9). In the simulations, for particles with a certain diameter (and larger), this supersaturation is reached for each path in the saturator. Thus, the calculated activation efficiency η_{a} cannot solely reflect the decreasing plateau counting efficiency η_{plat} with decreasing pressure. Additionally, the edge steepness ε of the calculated activation efficiency η_{a} stays almost the same for all pressure stages. Combined with the sampling efficiency η_{s} (Fig. 7), which takes the particle losses inside the inlet and saturator of the CPC into account, some of the decreasing plateau counting efficiency as well as some of the decreasing edge steepness ε (curves get flatter) can be explained. For the Grimm CPC, the onset diameter is also slightly shifted by the particle losses. Comparing the parameters of the simulations (Table 3) with those of the measurements (Table 2), the onset diameters d_{p,0} and the cutoff d_{p,50} for the TSI CPC are mostly within 0.5 nm. In general, the calculated counting efficiency (${\mathit{\eta}}_{\mathrm{calc}}={\mathit{\eta}}_{\mathrm{a}}\cdot {\mathit{\eta}}_{\mathrm{s}}$) captures the pressuredependent behavior of both CPC models quite well, except for the strong decline of the plateau counting efficiency η_{plat} at the lowest pressure stage.
For the pressuredependent shift of the plateau counting efficiency η_{plat}, other factors should also be considered. The losses inside the reference instrument, the FCE, influence the results of the measurements, but they are not included in the simulations (this can best be seen by the difference of η_{plat} (Table 2) and η(30 nm) (Table 3) at 1000 hPa). Another important factor for η_{plat} might be the losses during growth of activated particles and the losses inside the optics, which are summarized in the detector efficiency η_{d}. This detector efficiency η_{d} also includes the focusing of the particles into the optics, which was partly investigated by Takegawa and Sakurai (2011). Further research with measurements and (more advanced) simulations is needed to evaluate the pressure dependence of the detector efficiency η_{d}, which is beyond the scope of this publication.
In this study, we evaluated the pressuredependent performance of two CEN/TS 16976:2016specified CPC models, the Grimm 5410 CEN and the TSI 3772 CEN. The performance of four Grimm CPCs and two TSI CPCs was analyzed at 1000, 750, 500, and 250 hPa. Additionally, we added measurements at 375 hPa for the TSI CPCs and at 150 hPa for the Grimm CPCs. In general, we found a similar trend for the pressuredependent performance to that shown in other publications (e.g., Takegawa and Sakurai, 2011; Zhang and Liu, 1991; Bezantakos and Biskos, 2021; Mei et al., 2021): with decreasing pressure, the plateau counting efficiency η_{plat} is decreasing, the cutoff diameter d_{p,50} is increasing, and the edge steepness ε is decreasing. At 1000 and 750 hPa, both CPC models fulfill the CEN/TS 16976:2016 criteria. Below 500 hPa, the pressuredependent performance differs between the two CPC models; the Grimm CPCs have a higher plateau counting efficiency η_{plat} and a lower cutoff diameter d_{p,50} than the TSI CPCs. The onset diameter d_{p,0} stays almost constant for the Grimm CPCs.
To gain more insights into the different performance of the two CPC models, we conducted a simulation study. We simulated the temperature and vapor pressure profile inside the insulator and condenser of both CPC models. From this we calculated the activation efficiency η_{a} and combined it with the sampling efficiency η_{s}, which includes the diffusional losses inside the CPC. The simulation results capture the overall pressure dependence of the experimental results. The simulations reveal that for the TSI CPCs, the onset diameter d_{p,0}, and hence the counting efficiency curves, shifts due to the reduction of the saturation ratio, whereas for the Grimm CPCs, it mainly shifts because of diffusional losses. In addition, the decrease in the edge steepness ε with decreasing pressure can only be explained by including the sampling efficiency η_{s}. We have not included the detector efficiency η_{d} in our numerical study due to its complexity. To fully understand the pressure dependence of the counting efficiency, it is necessary to investigate the detector efficiency η_{d} further, which is beyond the scope of this study.
There are several approaches to eliminating the pressure dependence of CPCs by changing the design (e.g., Williamson et al., 2018; Wilson et al., 1983) or the working fluid (e.g., Hermann et al., 2005; Williamson et al., 2018) of the CPC or by altering the saturator and condenser temperatures (e.g., Hermann and Wiedensohler, 2001; Bezantakos and Biskos, 2021). Another approach to compensate for the pressuredependent effects might be to utilize the implications of Eq. (8). If the volumetric flow inside the CPCs is increased by the same factor as the pressure is reduced, the temperature and vapor pressure profile inside the insulator and condenser should stay the same. This would result in an activation efficiency η_{a} independent of the pressure (Zhang and Liu, 1991). However, the focusing of the particles into the laser beam and the optical counting is affected by the change in the volumetric flow rate. Therefore, the varying flow rate approach is not straightforward, but it might be interesting for further investigations.
For harmonizing the data from highalpine measurement stations (ACTRIS, etc.) or aircraft measurements, it might be of interest to include specifications on the pressuredependent performance of CPCs in a new standard. Here our approach to separate the different efficiencies and the results of our measurements could help to correct for the pressuredependent effects. Our findings might also be helpful for designing new CPCs.
Symbol  Parameter 
η_{CPC}  CPC counting efficiency 
η_{s}, η_{a}, η_{d}  Sampling, activation, and detector efficiency 
${\mathit{\eta}}_{\mathrm{CPC}}^{\prime}$  Measured counting efficiency 
η_{plat}  Plateau counting efficiency 
d_{p,50}  Cutoff diameter 
d_{p,0}  Onset diameter 
d_{p,50fit}  Fitted cutoff diameter 
ε  Edge steepness parameter $\mathit{\epsilon}=\mathrm{\Delta}\mathit{\eta}\left({d}_{\mathrm{p}}\right)/\mathrm{\Delta}{d}_{\mathrm{p}}$ 
T, T_{sat}, T_{con}, T_{wall}  Temperatures, saturator, condenser, and wall temperature 
Q, Q_{a}, Q_{s}  Flow, aerosol flow, and sample flow 
z, r, z^{′}, r^{′}  Axial z and radial r direction inside a tube; normalized axial ${z}^{\prime}=z/{R}_{\mathrm{t}}$ and radial ${r}^{\prime}=r/{R}_{\mathrm{t}}$ direction 
R_{t}  Radius of the tube (either condenser, insulation, or saturator tube radius) 
p_{v}, p_{sat}  Partial and saturation vapor pressure 
S  Saturation ratio $S={p}_{\mathrm{v}}/{p}_{\mathrm{sat}}$ 
Pe, Re, Pr, Sc  Péclet, Reynolds, Prandtl, and Schmidt number (dimensionless) 
D_{K,eq}  Equilibrium Kelvin diameter 
Code is available at https://www.doi.org/10.25365/phaidra.430 (Spät and Bauer, 2023).
Data are available at https://www.doi.org/10.25365/phaidra.427 (Bauer et al., 2023).
The supplement related to this article is available online at: https://doi.org/10.5194/amt1644452023supplement.
PSB and BW designed and supervised the study. PSB, DS, and ME prepared the experiment, tested the experimental setup, and performed the measurements. The experimental results are partly presented in the master's thesis of ME. AG developed the computer code and supporting algorithms for the instrumentation. PSB and DS analyzed the data with support from ME. DS performed the simulations with supervision and contributions from PSB. The interpretation of the results of the simulations is the outcome of numerous discussions among PSB, DS, and all coauthors. PSB wrote the manuscript with revisions from DS and BW. All coauthors read and commented on the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors acknowledge the great support during our measurement campaign from Gerhard Steiner (Grimm), Sebastian Schmitt (TSI), and Torsten Tritscher (TSI). The authors thank Tristan Reinisch (AVL DiTEST) for the discussion on the simulations of CPCs. Many thanks are given to Peter Wlasits and Paul Winkler (both University of Vienna) for the great discussions and their support and help with the instrumentation.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation framework program under grant agreement no. 640458 (ALIFE) and the ESA project ACARE (ESA contract no. 4000125810/18/NL/CT/gp).
The CPCs were purchased under the University of Vienna's investment program (IP734013) and through further support by the University of Vienna.
Openaccess funding was provided by the University of Vienna.
This paper was edited by Joachim Curtius and reviewed by Alfred Wiedensohler and one anonymous referee.
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 Abstract
 Introduction
 Methods
 Results and discussion
 Conclusion
 Appendix A: List of symbols – most important symbols in order of occurrence
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement
 Abstract
 Introduction
 Methods
 Results and discussion
 Conclusion
 Appendix A: List of symbols – most important symbols in order of occurrence
 Code availability
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Supplement